Consequently, if one were to set $v_i = 0$ for all outcomes, then the standard linear regression model and the mixed-effects meta-regression model are actually identical (with $\sigma^2$ denoting the same parameter as $\tau^2$). This equivalence can be demonstrated with an arbitrary dataset:

See help(stackloss) for more details on this dataset. Most importantly, variable stack.loss is the dependent variable with Air.Flow, Water.Temp, and Acid.Conc serving as potentially relevant predictors.

Now let's fit the standard linear regression model to these data with both the lm() and the rma() functions with:

Note that all sampling variances are set to 0 for rma() (the function will actually issue a warning that the dataset includes outcomes with non-positive sampling variances – which would be rather strange in the meta-analytic context – but this can be safely ignored here).

The estimated model coefficients, corresponding standard errors, and the test statistics are exactly the same. However, lm() computes the p-values based on the t-distribution, while rma() uses (by default) the standard normal distribution.

The $I^2$ and $H^2$ statistics typically reported by the rma() function are missing, since these statistics cannot be computed when the dataset includes outcomes with non-positive sampling variances. Similarly, the usually reported results from the $Q$-test for heterogeneity are omitted for the same reason.

To get full correspondence between the two models, we can use the 'Knapp & Hartung' method when fitting the model with the rma() function: